Computational model of integrin adhesion elongation under an actin fiber

Cells create physical connections with the extracellular environment through adhesions. Nascent adhesions form at the leading edge of migrating cells and either undergo cycles of disassembly and reassembly, or elongate and stabilize at the end of actin fibers. How adhesions assemble has been addressed in several studies, but the exact role of actin fibers in the elongation and stabilization of nascent adhesions remains largely elusive. To address this question, here we extended our computational model of adhesion assembly by incorporating an actin fiber that locally promotes integrin activation. The model revealed that an actin fiber promotes adhesion stabilization and elongation. Actomyosin contractility from the fiber also promotes adhesion stabilization and elongation, by strengthening integrin-ligand interactions, but only up to a force threshold. Above this force threshold, most integrin-ligand bonds fail, and the adhesion disassembles. In the absence of contraction, actin fibers still support adhesions stabilization. Collectively, our results provide a picture in which myosin activity is dispensable for adhesion stabilization and elongation under an actin fiber, offering a framework for interpreting several previous experimental observations.


Introduction
The ability of cells to form adhesions with the extracellular environment is critical to many physiological and pathological processes, including wound healing, tissue morphogenesis, embryonic development, and cancer metastasis [1][2][3]. Adhesions are hierarchical assemblies of~150 proteins that associate sequentially on the cell membrane, starting from transmembrane integrin receptors undergoing activation and connecting extracellular matrix (ECM) ligands to the actin cytoskeleton [4][5][6]. During cell spreading and migration, nascent adhesions form in a thin region at the cell protruding edge, called the lamellipodium. In the lamellipodium, actin filaments promote inside-out integrin activation and sustain edge motion [7][8][9][10][11]. Nascent adhesions either rapidly turn over or stabilize to the ECM near the base of the lamellipodium, where the actin filaments change their architecture from branched to bundled and form radial fibers, under which the adhesions elongate. Nascent adhesions under the actin fibers can further grow into focal complexes and mature adhesions through the recruitment of intracellular accessory proteins, such as talin, vinculin, paxillin, and zyxin, among others [12][13][14][15][16][17]. It is well established that the maturation of integrin-based adhesions involves the sequential recruitment of several accessory proteins, actomyosin contractility and intracellular signaling [18][19][20][21], but how nascent adhesions elongate at the end of actin fibers remains largely elusive.
Nascent adhesions appear in the lamellipodium as clusters containing an average of 20-50 integrins, with a diameter of~100 nm, smaller than the~0.25 μm of the light microscope [4,22]. They typically disassemble within seconds or minutes [4]. When they do not disassemble, they transition into elongated morphologies and connect with actin filaments in stress fibers. In these adhesions, individual integrins are highly dynamic and undergo cycles of ligand binding and unbinding, governed by inside-out activation through talin binding to the integrin cytoplasmic tails [11,[22][23][24].
Actomyosin contractility from the actin fibers stabilizes nascent adhesions and supports integrin connections with the ECM through reinforcement of their bonds with ECM ligands. Experiments with myosin inhibitors and dominant negative mutants have shown that actomyosin force also promotes the recruitment of intracellular accessory proteins and the maturation of adhesions [15,[25][26][27][28]. However, several experiments have demonstrated that adhesion elongation does not require myosin activity [29,30], which suggests that a force-independent stabilization of the nascent adhesions may precede myosin-mediated reinforcement. Without myosin activity, substrate stiffness is needed to support the stabilization of adhesions [31]. However, how exactly adhesions elongate and stabilize under an actin fiber is unknown.
In this study, we sought to understand the mechanisms of adhesion elongation and stabilization at the end of an actin fiber by testing the contributions of actomyosin force, inside-out activation, and substrate stiffness to these mechanisms. Because the transient nature of nascent adhesions makes it difficult to distinguish adhesion assembly from elongation and stabilization, here we applied computational modeling (a schematic of the model is provided in Fig 1).
Our results revealed that an increase in the fiber contractile activity alone stabilizes the adhesion up to a certain force threshold. Above this force threshold, most integrin-ligand bonds fail, and the adhesion disassembles. In the absence of contraction, inside-out activation of integrins under an actin fiber accounts for adhesion stabilization and elongation. Taken together, the results from our model demonstrate a myosin-independent mechanism for adhesion elongation, which is regulated by inside-out integrin activation.

Overview of the computational model
We extended our computational model of adhesion assembly based on overdamped Langevin dynamics [29,32] by incorporating the effect of an actin fiber. Our simulation domain was simplified to a 2D square of 1 μm side (Fig 1A). Ligands were fixed at the vertices of a 20x20 lattice. Integrins were represented as point particles that diffuse and transition between inactive to active states, with defined probabilities. When they were active and in proximity of a free ligand, integrins could establish harmonic interaction potentials with that ligand and become ligated. The duration of each bond was determined by the probability of rupture, k off , which was calculated from the total force on the bond. In particular, the lifetime (τ = 1/k off ) of the integrin-ligand bonds followed catch-slip kinetics, in which an initial strengthening of the bond occurred with increasing force, followed by weakening at higher forces ( Fig 1B). The effect of an actin fiber was incorporated in the model by Schematics of the 2D computational model. The domain consists of a grid of ideal springs (gray particles) with stiffness k. Integrins diffuse (green particles) with diffusion coefficient D and establish harmonic interactions with the substrate springs (magenta particles). If integrins are in the region of the actin fiber (pink area) a force, F myo is also added. F myo build tension on the integrin-ligand bond. This tension determines the bond lifetime, τ, which is the reciprocal of the rupture rate, k off , as τ = k off . B. The bond lifetime versus force relation for integrin-ligand bonds follows catch-slip bond kinetics.
https://doi.org/10.1371/journal.pcbi.1011237.g001 considering a stripe of 300 nm width, at the center of the domain (Fig 1A). The fiber locally changed the kinetic of integrins underneath it. Under the fiber, integrin activation was increased, depending on the degree of bundling of actin filaments, and a force (actomyosin force in the fiber, F myo ) was applied to all ligated integrins. Depending on the spring constant of the harmonic interaction between integrin and ligand, which mimicked substrate stiffness and contributed to the substrate force, F sub , and depending upon the magnitude of F myo in the fiber region, the total force on the integrin-ligand bonds was calculated and the probability of rupture determined. Activation, ligand binding and unbinding resulted in cycles of integrin diffusion, activation, and de-activation, which mimicked nascent adhesions. The main objective of the model was to test the effect of an actin fiber on the orientation and stability of the adhesion. The effects that fiber contractility had on several adhesion properties were evaluated by running simulations in the presence and absence of contraction, and using varying F myo . The effect of substrate stiffness, which further contributed to the total force on the integrin-ligand bonds, was also assessed.

Integrin adhesions are elongated under an actin fiber
We confirmed that integrin adhesions are elongated under actin fibers using COS7 kidney epithelial cells. Cells were co-transfected with mScarlet-paxillin to label adhesions and EGFP-F-tractin to label actin. Imaging by TIRF showed adhesions assembling at the advancing leading edge (Fig 2A), and larger elongated adhesions were found only back from the edge and under a thick actin fiber (Fig 2A and 2B). These data suggest that actin fibers have a role in the elongation of integrin adhesions at the back of the leading edge, consistent with previous reports [2,33,34].

Actomyosin contractility and substrate stiffness interplay in the assembly of adhesions
Since actomyosin force and ECM stiffness play an important role in the maturation of integrin adhesions [18,[34][35][36], we tested how actomyosin force from fibers can account for the stabilization of nascent adhesions. We ran simulations by systematically varying the magnitude of actomyosin force, F myo , up to 40 pN, and substrate stiffness, k, from 0.1 to 0.7 pN/nm. The average percentage of integrin-ligand bonds first increased with increasing F myo , then it decreased, and this effect was more marked at high substrate rigidities ( Fig 3A). For F myo = 15-30 pN, an increase in ligated integrins was observed with increasing k, which corresponded to an increase of the total time spent by integrins in the ligated state ( Fig 3B). A maximum of5 5-65% of ligated integrins was reached for F myo = 15-30 pN and k = 0.6-0.7 pN/nm (Fig 3A). A percentage of 55-65% corresponded to a minimum separation between ligated integrins of 70 nm, corresponding to the maximum interligand distance for adhesion stabilization [37,38], thus indicating a high probability of adhesion stabilization. For F myo > 30 pN, the percentage of ligated integrins dropped below 20% at all substrate rigidities, indicating low probability of adhesion stabilization. Because the activation rate of integrins was fixed, while their unbinding rate depended on the force on the integrin-ligand bonds (Fig 1B), the amount of unbinding was different for different values of F myo and k. For F myo = 15-30 pN, the rate of unbinding decreased with increasing substrate stiffness from k = 0.1 pN/nm to k = 0.7 pN/nm ( Fig 3C). The reduced unbinding of integrins resulted from the strengthening phase of the catch-slip bond, which emerged from the combined effect of actomyosin contractility and substrate stiffness ( Fig 1B). For F myo > 30 pN, unbinding increased on all substrates, because of the weakening phase of the catch-slip bond ( Fig 1B).
Together, these results indicated that actomyosin contractility and substrate stiffness need to be finely tuned to stabilize adhesions by reducing integrin unbinding. Increasing actomyosin contractility up to 30 pN, which is the force corresponding to the maximum lifetime of the catch-slip bond (Fig 1B), maximally reduces the unbinding of integrins. Increasing substrate stiffness augments this effect. When actomyosin contractility is higher than 30 pN the integrin-ligand bonds fail (Fig 3A and 3C), and the adhesion is unstable on all substrates.

Actin fibers supports elongation of adhesions
Since results from our model showed that actomyosin contractility first lead to adhesion stabilization, but then lead to adhesion failure (Fig 3A and 3C), we tested the hypothesis that actin filaments in a fiber, by promoting inside-out integrin activation, stabilize nascent adhesions. By comparing simulations without and with an actin fiber, we found that ligated integrins concentrate under the fiber (Fig 4A and 4B). The higher density of ligated integrins under the fiber was reflected in a higher total percentage of ligated integrins (Fig 4C). Like the case without an actin fiber (Fig 3A), incorporation of actomyosin contractility further increased the percentage of ligated integrins up to a threshold force above which bond failure occurred and the percentage of ligated integrins decreased (S1A Fig  Average percentage of ligated integrins using P bundling = 0 (no fiber) and P bundling = 1 (fiber). Errorbars indicate standard deviation from the mean. D. Average angle of the adhesion, relative to the direction of the actin fiber, using P bundling = 0 (no fiber) and P bundling = 1 (fiber). The angle is calculated from the direction of the first principal component of the 2D positions of ligated integrins, computed at each second of simulations between 100-500 s. Errorbars indicate standard deviation from the mean. E. Frequency of binding events using P bundling = 0 (no fiber) and P bundling = 1 (fiber). F. Distribution of total time spent by integrins in the ligated state, calculated as sum of the ligand-bound lifetimes of for each integrin over the course of 500 s of simulations, using P bundling = 0 (no fiber) and P bundling = 1 (fiber). All data are computed in the absence of actomyosin contractility, from 3 independent runs using k = 0.6 pN/nm. https://doi.org/10.1371/journal.pcbi.1011237.g004 the absence of contraction, the rate of ligand binding increased from~18 s -1 to about~23 s -1 (Fig 4E), which increased the total time spent by integrins in the ligated state by~50% ( Fig  4F). Increasing the width of the actin fiber increased the percentage of ligated integrins in the adhesion (S3A Fig Taken together, our results show that an actin fiber, by promoting integrin activation and without contraction, can increase integrin binding of substrate ligands, enhance the lifetime of integrin-ligand bonds, and promote the elongation of the adhesion along the fiber. These results can explain how adhesions change at the base of the lamellipodium, where the actin filaments change architecture from branched to bundled and contractility is not significant.

Interplay between actin fiber contractility and bundling on adhesion stabilization
The results from our model showed that an actin fiber can support the elongation and stabilization of an integrin adhesion in the absence of contraction (Fig 4). Since actin fibers are typically contractile and present varying numbers of crosslinking proteins and motors [39][40][41], we examined how contractility and actin bundling affects adhesions. We assessed how systematic variations in the probabilities of actin bundling, P bundling , and actomyosin contractility, F myo , affect the density of ligand-bound integrins, the time that integrins spent in the ligated state, and the orientation of the adhesion. By increasing P bundling , the average percentage of ligated integrins increased from~25%, up to~50%, depending on F myo (Fig 5A). Without an actin fiber (P bundling = 0), a four-fold increase in F myo , from 5 pN to 20 pN, increased the average percentage of ligated integrins from~25% to~35%, a total increase of about 40% (Fig 5A). Using low actomyosin contractility, F myo = 5 pN, increasing P bundling from 0 to 1 increased the average percentage of ligand-bound integrins of the same amount ( Fig 5A). With F myo = 20 pN, varying P bundling from 0 to 1 increased the fraction of ligand-bound integrins from~35% to~45%, corresponding to a total increase of~30%. However, with higher actomyosin force, F myo = 30 pN, the same increase in P bundling increased the fraction of ligated integrin from 30% to only~35%, a total increase of~16%. These results indicated that, like the effect of contractility, actin filaments bundling in the fiber also supports adhesion stabilization. When acting together, bundling augment the effect of contractility to stabilize adhesions. However, when actomyosin contractility is high, the effect from bundling on adhesion stabilization is reduced. Therefore, actin bundling alone or with intermediate contractility maximally supports adhesion stabilization. The observed increases in the fractions of ligated integrins as a function of P bundling originated from the increased activation rate of integrins under the fiber, resulting in a longer time that integrins spent in the ligated state ( Fig 5B). For F myo = 5 pN, increasing P bundling from 0 to 1 increased the total bound time of~40% (Fig 5B), an amount comparable to the corresponding variation in the percentage of ligand-bound integrins under the same conditions ( Fig 5A). For F myo = 20 pN, the same increase in P bundling increased the total time spent by integrins in the bound state of~15% (Fig 5B), again comparable to the increase in percentage of ligated integrins under the same conditions ( Fig 5A). Increasing substrate stiffness together with F myo stabilized the adhesion at all levels of P bundling (Fig 5C), with the more marked effect occurring when P bundling = 1. At k = 0.6-0.7 pN/nm and F myo = 20 pN, the maximum value of ligated integrins increased from~35% at P bundling = 0, to~40% using P bundling = 0.5, and~45% using P bundling = 1 (Fig 5C). Plots of the distribution of the tension before bond breakage showed comparable average values between absence and presence of bundling but increases in the spread of the distribution with increasing F myo using P bundling = 1 ( Fig 5D). Actin bundling also affected the orientation of the adhesion, by promoting its alignment along the fiber on all substrates ( Fig 5E).
Collectively, results from our model demonstrate that an actin fiber supports adhesion elongation on all substrates and augment the effect of contractility in stabilizing adhesions.

Discussion
Nascent adhesions form in the lamellipodium of adherent cells and elongate at the end of actin fibers radiating perpendicularly to the cell edge [4]. Experiments probing the mechanisms of adhesion elongation and stabilization at the end of actin fibers have variably suggested a requirement for contractile force [3,28,[42][43][44] versus bundling of the actin filaments in the fiber [4,45]. As a result, the mechanisms governing elongation and stabilization of adhesions are not well established. In this study, we extended our computational model of integrin adhesion assembly based on Brownian dynamics [29,32] to test the contributions of actomyosin contractility and actin bundling to the dynamics of adhesions. Results from this study revealed that an actin fiber, by providing a physical template for inside-out integrin activation, guides the elongation of nascent adhesions in the direction of the fiber (Figs 4 and 5). Forces from actomyosin contractility also promote adhesion stabilization and elongation, but only up to a certain force level, then they lead to adhesion failure and disassembly (Fig 3A and 3B).
Brownian dynamics simulation approaches are widely used for understanding the dynamics of biological macromolecules. Our computational model used this method to reproduce cycles of free diffusion and immobilization of integrins to the ECM. Together with the thermal effects on integrin motion, that governed integrin diffusion, our model also evaluated kinetic rate constants for ligand binding and unbinding, which determined transitions between free and ligated integrins. Displacements of integrins were calculated over time, based on the sum of forces acting on them. When integrins were bound to substate ligands, they were subjected to the force from actomyosin contractility in the fiber region, and the substrate resisting deformation. Force from the substrate varied proportionally with substrate stiffness, k, and modulated the rate of bond rupture, k off . k off decreased with increasing k, so that integrins unbound less easily when k was high, resulting in an increased average fraction of ligated integrins on stiff substrates (Fig 3). Bundling was incorporated by increasing the rate of integrin activation under the fiber. Variations in substrate stiffness were incorporated using different spring constants for integrin-ligand bonds, as in previous models [46][47][48]. Thus, the force on the i-th ligated integrin, F i , was the sum of: (i) stochastic force from thermal effects, F T ; (ii) actomyosin contractility in the region of the actin fiber, F myo = 5-40 pN, a range comparable to the contractility of actomyosin bundles in cells [49,50]; and (iii) force from the substrate limiting integrin motion away from the ligand, F sub , proportional to the substrate stiffness k (between 0.1-0.7 pN/nm [46,51]). Considering only the effects of thermal force on integrin motion, the total force on ligated integrins, F i , was in the single pN range, comparable to the average force/ integrin reported for single talin molecules [52]. In the presence of actomyosin contractility in the fiber region, the force on ligated integrins reached values of few tens of pN, consistent with experimental measurements of maximal force per integrin, up to~40 pN [53,54].
Like our previous study of nascent adhesion assembly and sensing of substrate stiffness [29], results from our model showed that substrate stiffness can account for adhesion stabilization (Fig 3A and 3B). With increasing F myo between 5-40 pN, adhesions initially stabilized, reaching an optimum at F myo = 25 pN, but then destabilized for F myo > 30 pN (Fig 3A and 3B). As the load on integrins increased, the lifetime of the integrin-ligand bonds first increased, and then decreased (Fig 1B), following catch-slip bond kinetics [55,56]. When individual ligand-bound integrins were subjected to forces above 30 pN, the decrease in bond lifetime led to bond failure. Soft substrates did not support adhesions, resulting in reduction of the density of ligated integrins (Fig 3A). Previous studies have shown that cell contraction is not needed for the assembly of adhesions [29]. However, previous studies have also shown that in the absence of an external force, such as substrate stiffness, actomyosin force promotes adhesion stabilization [31]. This result is consistent with our model result that fiber contractility promotes adhesion stabilization on softer substrates (Fig 3).
Our model also demonstrated a critical role for actin bundling in adhesion elongation and stabilization (Fig 4). Actin bundling increased integrin immobilization under the fiber (Fig 4A  and 4B), the average density of ligated integrins in the adhesion (Fig 4C) and the alignment of the adhesion with the fiber (Fig 4D). Alignment of the adhesion with the fiber also correlated with the width of the fiber (S1C Fig). The increase in the percentage of ligated integrins from actin bundling resulted in a reduction of their spatial separation, a prerequisite for adhesion stabilization [37,38]. Actomyosin force and substrate stiffness affected the rupture rate of integrins through modulation of their lifetimes (Fig 1B). We have previously shown that increasing the concentrations of motors and crosslinking proteins in an actin fiber increases the fiber force, but without crosslinkers high forces cannot be sustained, despite the crosslinking activity of the motors [41]. In cells, force generation from motors and motor crosslinking overlap. However, several previous studies implicated myosin crosslinking rather than contractility as important in adhesion elongation [4,45], which supports our model results that bundling in the absence of contractility stabilize the adhesion (Fig 4). In CHO.K1 ovary epithelial-like cells, a myosin mutant that retained actin cross-linking activity but lacked motor activity rescued adhesion elongation in myosin IIA-null cells [4]. Similarly, in U2OS human osteosarcoma cells, actin bundling was sufficient to rescue the inhibition of myosin activity to drive adhesion elongation and increase the lifetime [30,34,45]. Results from our model measured the lifetime and the total bound time of individual integrin-ligand bonds and not the lifetime of the whole adhesion. However, with increasing lifetime of individual ligand-bonds, the adhesion lifetime also increases. Therefore, our results are qualitatively consistent with these previous experiments and collectively support a picture in which actin bundling can alone support adhesion elongation in the absence of contraction (Fig 4).
The impact of actin filaments on nascent adhesions from this model also provides a new framework to understand actin-mediated nucleation of nascent adhesions in the lamellipodium. Previous studies have indicated that actin polymerization drives clusters of activated integrins probing for sites of adhesion nucleation [57]. Similarly, Arp2/3-mediated actin branching around integrin sites has been shown to have an important role in the formation of nascent adhesions [58][59][60]. Our model suggests a mechanism by which actin-mediated inside-out activation of integrins increases the density of ligated integrins under the fiber, and therefore co-localization of filaments and integrins promote adhesion assembly and stability.
Our model assumes that ligated integrins are connected to the actin cytoskeleton through talin [61]. Talin was considered implicitly by applying actin flow and F myo to ligated integrins. Because the cytoskeletal force was directly transmitted to the integrin-ligand bonds, this force was not buffered by talin unfolding, and vinculin binding to talin was not incorporated. Extensions of this model to incorporate these effects will be considered in future studies. In the current implementation, our model demonstrates that nascent adhesions can elongate in the direction of an actin fiber without talin unfolding, force redistribution and vinculin binding. This supports the idea that, provided that an actin fiber works as a physical template for integrin activation, talin unfolding and vinculin recruitment are dispensable for adhesion elongation. This result is consistent with experimental studies showing that: initial cell spreading only requires integrin-ligand bonds and that nascent adhesions elongate in the absence of vinculin [4,58].
The results from our model are consistent with a myosin-independent mechanism for adhesion elongation. We suggest that the initial phases of elongation of integrin adhesions emerge passively from the actin fiber acting as a template for integrin activation in the absence of other reinforcement mechanisms. It is plausible that the recruitment of mechano-sensitive intracellular proteins stabilizes integrin-ligand bonds on stiff substrates.
Since contractile myosin activity correlates with the maturation of elongated adhesions in the lamella [20,25,62] and has been presumed to be the important mechanism for adhesion stabilization [28,34,63], it remains plausible that myosin-dependent contractility stabilizes nascent adhesions into focal adhesions and recruits' proteins for specialized non-lamellipodium functions. Contractility may be involved in adhesion maturation into large adhesions in the cell middle or adhesions with different compositions in scenarios specific to substrate signaling or mechanical stretch.

Materials and methods
To elucidate how nascent adhesions elongate under actin fibers, we extended our Brownian dynamics model of integrin adhesions assembly [29,32]. An actin fiber was incorporated implicitly in the model, as a region of space where actin filaments bundling and actomyosin contractility affect integrin dynamics. We used the model to evaluate how systematic variations in the probability of actin filaments bundling and in the magnitude of the actomyosin force affect the percentage of ligand-bound integrins, the total time that integrins remain ligated and the orientation of the adhesion.

Simulation domain
The simulation domain is a 2D square of 1 μm side, which corresponds to an area of cell membrane where integrins bind ligands and form an adhesion (Fig 1A). The actin fiber corresponds to a central strip of the domain of 300 nm width, in which bundling and actomyosin force control integrin dynamics. Integrins are initially randomly distributed in the domain and undergo free diffusion. Periodic boundary conditions are used to avoid finite boundary effects on integrin motion. Ligands are represented as fixed nodes on a lattice of 20x20 cells (each cell has sides of 50 nm). Interactions between integrins and ligands are governed by kinetic rates and an adhesion is considered formed when multiple bonds between integrins and ligands exist in the domain.

Integrin and ligand representations
The model considers 300 integrins and 441 immobilized ligands. Each i-th integrin and j-th ligand are defined by a 2D position vectors, r i and r j , respectively. The vector r i presents x, and y coordinates of the i-th integrin; the vector r j presented x, and y coordinates of the j-th ligand. At every timestep of the simulations, components x and y of r i are updated to track integrin displacement, while components x, and y of r j remain fixed.

Brownian Dynamics simulations via Langevin equation
Recognizing that inertia is negligible on the length and time scales of integrin motion in the plasma membrane, the displacement of each i-th integrin depends on the total force on it, F i , and is governed by the Langevin equation of motion in the limit of high friction [64]: where r i is the position vector; z i is the friction coefficient equal to 0.071 pN s/μm, which corresponds to a diffusion coefficient D = 0.058 μm 2 /s [23], obtained from Einstein relation as where k B T = 4.11 pN nm. The explicit Euler integration scheme is used to displace integrins depending on F i , z i and dt, as:

Forces on integrins
The total force on each i-th integrin, F i , includes a stochastic contribution from thermal effects, F T , and a deterministic contribution from: substrate stiffness, F sub , and, in the fiber region, actomyosin contractility, F myo . It is calculated as: Thermal force, substrate force, and actomyosin contractility are all calculated at each timestep of the simulations and used to update integrin positions (Eqs 1 and 2). When integrins are diffusing, only F T acts on them, while F sub = 0 and F myo = 0. When integrins are ligated, F sub is added. When integrins are ligated and under the actin fiber, F myo is also considered.
Since actin flow was shown to govern a rearward movement of integrins in the adhesion, we tested the effects of actin flow velocity on the distribution of force on integrin-ligand bonds. We imposed a displacement of ligated integrins corresponding to experimental actin flows, up to 100 nm/s [8,65]. Our results showed that the distribution of force on ligated integrins was not affect by actin flow (S4 Fig). Therefore, our model does not include the contribution from the actin flow to the motion of integrins.

Stochastic force acting on integrins
A stochastic force, F T , satisfying the fluctuation-dissipation theorem, is applied to all integrins, to mimic thermal effects generating diffusion. F T has two force components, where each component is chosen from a Gaussian distribution with average 0, and standard deviation ffi ffi ffi ffi ffi ffi ffi ffi ffi At each timestep, the 2D displacement of the i-th integrin due to the thermal force is computed from the two force components, F T,x and F T,y , as: Therefore, at each dt the maximum displacement from thermal effects is~3.5 nm.

Deterministic forces acting on ligated integrins: Actomyosin force and substrate force
A deterministic force is applied on each ligated i-th integrin. It originates from actomyosin contractility in the fiber region, F myo , and substrate tension, F sub . F myo pushes integrins away from their ligand, in the horizontal direction (Fig 1A). F sub counters F myo by pulling ligated integrins towards the ligands, to restore the equilibrium distance, L.
F myo mimics contractile motors in the fiber and is applied on ligated integrins in the region where the fiber overlaps with the adhesion. We systematically varied F myo between 5-40 pN, which is a range typical of contractile actin bundles [49,50]. Without considering the opposing substrate stiffness, application of this force corresponds to a displacement Dr i ¼ The force from the substrate, F sub , is proportional to its stiffness, k, and follows Hookes' law, as: where ΔL is the deviation from L, and k = 0.1-0.7 pN/nm, as in previous models [46][47][48]. Considering x i and y i as coordinates of the i-integrin and x j and y j as coordinates of the bound j-th ligand, their separation relative to the equilibrium distance, L, is: DL ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi The displacements of ligand-bound integrin in each direction are calculated as:

Binding and unbinding of ligands
Integrins undergo cycles of free diffusion, binding, and unbinding of ligands. When a free, diffusive integrin comes in proximity of a free ligand (< 20 nm from it, which is a dimension characteristic of integrin headpiece extension [66]), it can bind the ligand by establishing an elastic interaction with probability: p on ¼ 1 À e À k on dt , where k on = 1 s -1 is the activation rate of integrin, of the same order of those used in [32], and dt = 0.0001 s is the timestep.
The bond between integrin and the ligand has spring constant, k, proportional to the substrate stiffness, and equilibrium distance L = 0.01 nm. The integrin-ligand bond lifetime, τ, depends on the total force on integrin, F i , and is the inverse of the rupture rate: τ = 1/k off . The rupture rate follows a catch-slip bond kinetics, in which the lifetime of the bond initially increases with force F i , then decreases, as shown in Fig 1B [ 55]. Following the Bell model [67], k off , is computed from a double exponential function, including a strengthening pathway represented by a first exponential term with a negative exponent; and weakening pathway, represented by a second exponential term with a positive exponent [68,69], as: Since the force on each bond is unique, depending on whether integrin is in the fiber region, and the magnitude and direction of F T , a specific τ exists for each bond. The rupture rate, k off , determines the probability of unbinding as: p off ¼ 1 À e À k off dt . Once integrins unbind their ligands, they become diffusive again until they bind a new ligand.

Actin fiber representation
In the model, the location where the actin fiber overlaps with the adhesion is incorporated as a 300 nm wide strip in the middle of the domain (Fig 1A). In this location, integrin activation is increased, to mimic cytoplasmic signals promoting integrin conformational extension and an increase in ligand binding affinity, consistent with experimental observations [9,10].
Actin bundling is incorporated in the code through its effect on integrin activation under the fiber. The probability of actin bundling, P bundling , is the probability that free integrins in the fiber area activate faster. Outside the fiber, integrins activate at a rate k on = 1 s -1 , as in our previous study [32]. When the probability of actin bundling is 0.5, about 50% of free integrins under the fiber present k on = 3 s -1 and the remaining 50% present k on = 1 s -1 . When the probability of actin bundling is 1, all integrins under the fiber present k on = 3 s -1 . The increase in rate of integrin activation under the fiber mimics talin-mediated activation of integrin [70]. Because experimental studies have also indicated that integrins under actin fibers undergo slow and directed movement along the fiber [57,71,72], we tested the effect of lowering the diffusion coefficient of free integrins in the fiber region, by increasing z i . We found that reducing diffusion increases adhesion alignment in the direction of the fiber, but to a smaller extent than increasing the activation rate ( S2C Fig).

Algorithm implementation
The total simulation time is between 300-500 s, which is a relevant time scale for nascent adhesions assembly and elongation [4]. Results are extracted after 100 s of simulations, when the number of ligated integrins reaches a steady value.
The algorithm consists of two parts: an initialization function and a step function. The initialization function sets the domain geometry and the boundary conditions (periodic in x and y) and assigns all simulation parameters to specific variables, including integrin diffusion constant, integrin and ligand concentrations and substrate stiffness; then, it sets the initial random positions of integrins and the positions of the ligands in the lattice. The step function runs iteratively, and each iteration corresponds to one timestep, until the total time of 300-500 s is reached. The main objective of the step function is to evaluate the total force on each integrin and update its position and bound state.

Initialization function
Parameter initialization includes the definition of domain geometry and size, number of integrins (300), and number of ligands (441). Integrins are initially distributed on the 1 x 1 μm domain, by assigning x and y coordinates randomly between -0.5 and 0.5 μm in both directions. The square domain is further divided into 20x20 cells, for a total of 400 cells, with ligands fixed at their vertices. Since the lateral domain size is 1 μm and there are 20 cells per side, neighboring ligands are separated by 50 nm, a value below the maximum ligand separation of 70 nm needed for adhesion stabilization [37,38].

Step function
The step function is divided into three parts: it first evaluates integrin positions; it computes the total force,  Distribution of the total time spent by integrins in the ligated state, in different conditions of F myo (between 0-30 pN), without (A) and with (B) actin bundling. C. Average angle of the adhesion relative to the actin fiber. Three conditions are tested: absence of bundling; bundling with reduced diffusion of integrin in the fiber region (using 5-fold higher z i than outside the fiber); and bundling with increased activation rate (using 3-fold higher k on relative to k on outside the fiber). The angle is calculated from the direction of the first principal component considering the 2D positions of ligated integrins. All data are computed between 100-500 s of simulations, using k = 0.6 pN/nm, and recording every 1 s. (TIFF)